Question 1106912
the price of the car is 12,000.
you have 8,000.
the annual interest rate is 3% compounded quarterly.
how many years before you can purchase the car?


the formula to use is f = p * (1 + r) ^ n


f is the future value.
p is the present value.
r is the interest rate per time period.
n is the number of time periods.


since you are compounding the interest rate quarterly, your interest rate per quarter is equal to 3% / 4 which is equal to .75% expressed as a percent and .0075 expressed as a rate.

in the formula, you use the rate, not the percent.


n is also expressed in quarters of a year, so if you want to know the number of years, you would have to divide n by 4 as well, once you find it.


your formula becomes 12,000 = 8,000 * (1 + .0075) ^ n.


divide both sides of this equation by 8,000 to get 12,000 / 8,000 = (1 + .0075) ^ n.


take the log of both sies of this equation to get log(12,000/8,000) = log(1.0075^n).


since log(1.0075^n) is equal to n * log(1.0075), your formula becomes log(12,000/8,000) = n * log(1.0075).


divide both sides of this equation by log(1.0075) and you get log(12,000/8,000) / log(1.0075) = n.


solve for n to get n  = 54.2644945.


replace n in the original equation with this to get 12,000 = 8,000 * (1.0075)^54.2644945.


the result is 12,000 = 12,000.


recall that n is in quarters, so divide 54.2644945 by 4 to get 13.56612362 years.


an alternate way of analyzing this is to use the formula f = p * (1 + r/c) ^ (n*c).


this formula assumes annual interest rate and number of years and incorporates the number of compounding periods per year into it.


using this formula, you would get 12,000 = 8,000 * (1 + .03/4) ^ (n*4).


you would ue the same procedure to get log(12,000 / 8,000) / log(1 + .03/4) = n*4.


you would solve for n * 4 to get n * 4 = 54.2644945.


you would then solve for n to get n = 13.56612362.


in the formula of f = p * (1 + r/c) ^ (n*c), .....


f = future value as before
p = present value as before
r = annual interest rate (not the annual interest rate percent).
n = number of years
c = number of compounding periods per year.


in the first formula of f = p * (1 + r) ^ n,
r is assumed to be per time period and n is assumed to be number of time periods.
the adjustment from years to time periods is done prior to using the formula.



in the second formula of f = p * (1 + r/c) ^ (n*c),
r is assumed to be per year and n is assumed to be number of years and c is the number of compounding periods per year.
the adjustment from years to time periods is incorporated as part of the formula.